There are certain receivers that are usable with, as an example, multiple input, multiple output (MIMO) code division multiple access (CDMA) signals. MIMO technology, that uses multiple antennas at both the transmitter and the receiver, has recently emerged as a significant breakthrough to increase spectral efficiency. Early efforts in this area are known as D-BLAST, see G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas”, Bell Labs Tech. J., pp. 41-59, 1996, and a more realistic strategy known as V-BLAST, see G. D. Golden, J. G. Foschini, R. A. Valenzuela, and P. W. Wolniansky, “Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture,” Electron. Lett., vol. 35, pp. 14-15, January 1999. To support multimedia services, UMTS and CDMA2000 extensions optimized for data services have lead to the standardization of Multi-Code CDMA systems-such as the High-Speed-Downlink-Packet-Access (HSDPA) and its equivalent 1×EV-DV (Evolution Data and Voice). Recently, MIMO extensions for the 3G wireless systems have received more and more attentions from the research community, as evidenced by A. Wiesel, L. García, J. Vidal, A. Pagès, Javier R. Fonollosa, “Turbo linear dispersion space time coding for MIMO HSDPA systems”, 12th IST Summit on Mobile and Wireless Communications, Jun. 15-18, 2003, Aveiro, Portugal.
However, originally the MIMO-based spatial multiplexing was proposed for narrow band and flat-fading channels. In a multipath-fading channel, the orthogonality of the spreading codes is destroyed and Multiple-Access-Interference (MAI), along with the Inter-Symbol-Interference (ISI), is introduced. The conventional Rake receiver does not provide satisfactory performance with a small spreading gain.
The LMMSE (Linear-Minimum-Mean-Square-Error)-based chip-level equalizer has the potential to restore the orthogonality of the spreading code, and to suppress both the ISI and MAI. Reference in this regard can be made to K. Hooli, M. Juntti, M. J. Heikkila, P. Komulainen, M. Latva-aho, J. Lilleberg, “Chip-level channel equalization in WCDMA downlink”, EURASIP Journal on Applied Signal Processing, August 2002, pp. 757-770, and to M. J. Heikkila, K. Ruotsalainen and J. Lilleberg, “Space-time equalization using conjugate-gradient algorithm in WCDMA downlink”, IEEE Proceeding in PIMRC, pp. 673-677, 2002. However, the use of the LMMSE equalizer involves the inverse of a large correlation matrix with a complexity at the order of O((NF)3), where N is the number of receive (Rx) antennas and F is the channel length. This can be prohibitively complex for realizing a real-time hardware implementation (see P. Radosavljevic, J. R. Cavallaro, A. D. Baynast, “Implementation of channel equalization for MIMO systems in WCDMA downlink”, submitted to ICASSP 2004, and Y. Guo, J. Zhang, D. McCain, J. R. Cavallaro, “Scalable FPGA architectures for LMMSE-based SIMO chip equalizer in HSDPA downlink”, 37th IEEE Asilomar Conference on Signals, Systems and Computers, 2003.
The fact that the MIMO CDMA receiver is to be embedded into a portable device makes the design of low complexity mobile receivers very critical for widespread commercial deployment of low cost products. To avoid the Direct-Matrix-Inverse (DMI), adaptive stochastic gradient algorithms such as LMS could be applied (see, in this regard, the above cited K. Hooli et al.) However, such adaptive stochastic gradient algorithms suffer from stability problems because the convergence depends on the choice of good step size.
Because the system is Hermitian and positive definite, a Conjugate Gradient algorithm has been proposed for iterative computation of the equalizer taps (see the above-cited M. J. Heikkila et al. and P. Radosavljevic et al.) The complexity of the CG is at the order of O((NF)2), which may be considered as a fast version algorithm. However, when multi-antenna receiver is applied, the signal dimension increases. The required MIMO equalizer filter length is high and the co-variance matrix has a very large eigen value spread. Although multiplication is not very expensive, and is relatively easy to implement on computers and is effectively parallelizable for structured matrices represented in compressed form, the structure rapidly deteriorates during the process of the iteration. The resulting complexity is still excessive for a hardware implementation (see Y. Guo, J. Zhang, D. McCain, J. Cavallaro, “Efficient MIMO equalization for downlink multi-code CDMA: complexity optimization and comparative study”, submitted to IEEE GlobeCom 2004.
In the following publications: V. Y. Pan, A. Zheng, “Superfast algorithms for Cauchy-like matrix computations and extensions”, Linear algebra and its applications, 310, 83-108, 2000, and V. Y. Pan, “Structured matrices and polynomials: unified superfast algorithms”, Springer, 2001, the authors presented superfast algorithms for structured matrices. For an N-dimension matrix, a superfast algorithm should have the complexity at the order of O(N log2(N)).
A FFT-based solution of the equalizer using circulant approximation is proposed by the above cited J. Zhang et al., and by the above-noted Y. Guo, J. Zhang, D. McCain, J. Cavallaro, “Efficient MIMO equalization for downlink multi-code CDMA: complexity optimization and comparative study”, submitted to IEEE GlobeCom 2004. However, circular corners need to be added to approximate the block Toeplitz structure with circulant structure. This may increase the condition number and reduce the system stability. In very high Signal-to-Noise-Ratio (SNR) and high Geometry range, the high condition number tends to degrade the system performance.